What makes e the natural base?: a maths EPQ idea
A title to start from
What makes e the 'natural' base for exponentials and logarithms?
Why it works as an EPQ
e appears from compound interest, from calculus and from series, and you can judge which of these explains 'natural' best.
Scope and difficulty
Accessible. Accessible. Three definitions, shown to agree, with a reasoned conclusion.
The maths
Builds on these A Level topics: Exponentials and logarithms · Differentiation · Sequences and series.
You would learn:
- Limits
- Power series
- Why the definitions agree
One possible plan
- Show (1 + 1/n)ⁿ approaching e with continuous compounding.
- Show eˣ is the exponential whose gradient equals its value.
- Sum the series for e and compare speeds of convergence.
- Decide which property most deserves the word 'natural'.
Pitfalls
- Asserting the definitions agree without showing why.
- Giving rounding errors as if they were evidence of convergence.
Where to start reading
- e: The Story of a Number (Eli Maor)
- Search for: equivalence of definitions of e
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